Matrix Jacobi

Both the matrices K and Kj. have to be regular. They are called Jacobi matrices. In general the Jacobi matrix for two linear independent steps of a reaction (s = 2) has the following form  

As mentioned above, the reduced degrees of advancement are used (without linear dependencies). can be obtained accordingly. For this reason, one can understand eqs. (2.31) and (2.35) as a development of x and a into a Taylor series to x and Aa at time t = 0, respectively. The differential equation (eq. (2.35)) can even be simplified. At time f — o the change in the degree of advancement has to become 

The relationship (2.39) is useful, since this equation allows the calculation of the values of the degrees of advancement at the end of the reaction using the initial slopes. This result can be compared to the measurement. Since the Jacobi matrix has to be regular, using eq. (2.38) 

40) and (2,41) are systems of s linear homogeneous differential equations with constant coefficients for all the s unknown and Aa,- respectively. These systems of equations are equivalent to a single differential equation of jth order for a single unknown [17]. Taking eq. (2.41), this differential equation for the tth concentration is given by 

In this differential equation the coefficients M, (i = l,2.s) represent the sums of the main minors [18] of the Jacobi matrix (for comparison see eq. (2.48)). 

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In this Eq. (Js)n is the Jacobi matrix for the solid phase, which contains the derivatives of the mass residuals for the particulate phase to the solid volume fraction. Explicit expressions for the elements of the Jacobi matrix can be obtained from the continuity for the solid phase and the momentum equations. For example for the central element, the following expression is obtained from the solid phase continuity equation, in which the convective terms are evaluated with central finite difference expressions ... 

Together with Eq. (93), this equation forms a set of equations from which explicit expressions for the derivatives of the velocities can readily be obtained. Expressions for the y- and z-direction and for the other elements of the Jacobi matrix are obtained in a similar manner. 

NEITHER THRESHOLD ATTAINED ESTIMATE OF THE SOLUTION FUNCTION VALUES AT THE FINAL ESTIMATE ESTIMATE OF THE INVERSE OF THE JACOBI MATRIX... 

Check it out. Linear algebra texts describe an analytical procedure using determinants, but computational chemistry employs a numerical iterative procedure called Jacobi matrix diagonalization, or some related method, in which the off-diagonal elements are made to approach zero. 

The Jacobi matrix and the Lanczos polynomial matrix within... 

Here, the elements U m are alternatively denoted by c0J , to indicate that the matrix U in the basis IVO is a tridiagonal matrix, which is equivalently called the Jacobi matrix or J-matrix and denoted by J = Hence, in the Lanczos basis, the evolution matrix U automatically acquires its tridiagonal (codiagonal) form of the type of a J-matrix in a finite dimension, say MxM such that UM = fi] = c0Jm- This is due to the definition of the basis... 

This is a band-structured matrix or a band matrix which is in this particular tridiagonal form also called the Jacobi matrix. Projecting both sides of Eq. (53) onto Pn+ fn+ I and using Eq. (59), we find ... 

This is an ordinary eigenvalue problem in which the tridiagonal Jacobi matrix Jxj is given in Eq. (60) with M = oo. The residues dk) are defined by Eq. (15), where IT ) is the exact complete state vector normalized to Co 0. The same type of definition for dk is valid for an approximation such as Eq. (67), provided that normalization is properly included according to Eq. (69) ... 

The Jacobi state vectors without diagonalization of the Jacobi matrix... 

This recurrence relation and the available triple set an, fin uk] are sufficient to completely determine the state vector Q without any diagonalization of the associated Jacobi matrix U = c0J, which is given in Eq. (60). Of course, diagonalization of J might be used to obtain the eigenvalues uk, but this is not the only approach at our disposal. Alternatively, the same set uk is also obtainable by rooting the characteristic polynomial or eigenpolynomial from... 

This is the orthogonality relation of the two Lanczos polynomials Q (m) and Qm(u) with the weight function, which is the residue dk [48]. We recall that the sequence Q = (Q ( z-) coincides with the set of eigenvectors of the Jacobi matrix (60). 

Determine the parameter values bj andZ>2 by using the data given in Example 9.1 and the nonlinear least squares method. Recall that in Example 9.1 we needed the elements of the Jacobian matrix 7 (see equation (9.142)). In this case, integrate simultaneously the time dependent sensitivity coefficients (i.e., the Jacobi matrix elements dyfdb and dyjdb2 ) and the differential equations. The needed three differential equations can be developed by taking the total derivative (as shown below) of the right hand side of equation (9.149) which we call h ... 

Equation (3.12) clearly illustrates that the roots of Pn(0, namely the nodes of the quadrature approximation a, are the eigenvalues of the tridiagonal matrix appearing in the equation. This matrix can be made symmetric (preserving the eigenvalues) by a diagonal similarity transformation to give a Jacobi matrix ... 

This procedure transforms the ill-conditioned problem of finding the roots of a polynomial into the well-conditioned problem of finding the eigenvalues and eigenvectors of a tridiagonal symmetric matrix. As shown by Wilf (1962), the N weights can then be calculated as Wa = OToV ai where tpai is the first component of the ath eigenvector (pa of the Jacobi matrix. 

After applying the PD algorithm the following Jacobi matrix is obtained ... 

Sack Donovan (1971) proposed an alternative approach for the calculation of the coefficient of the recursive formula reported in Eq. (3.5) (and appearing also in the Jacobi matrix) that resulted in higher stability. This approach is based on the idea of using a different set of basis functions naif) to represent the orthogonal polynomials, rather than the usual powers of f. The improved stability results from the ability of the new polynomial basis to better sample the integration interval. The coefficients are calculated from the modified moments defined as follows ... 

N - a - 1. Finally, the coefficients for the Jacobi matrix are computed as follows ... 

The fact that the Jacobi matrix is known can be used to evaluate very accurately integrals involving 6a (x,y) by applying Gaussian quadrature. For example, Eq. (3.84) leads to... 

Below a Matlab script for the calculation of a quadrature approximation of order N from a known set of moments iti using the Wheeler algorithm is reported. The script computes the intermediate coefficients sigma and the jacobi matrix, and, as for the PD algorithm, determines the nodes and weights of the quadrature approximation from the eigenvalues and eigenvectors of the matrix. 

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